Follow these steps to solve an absolute value equality
which contains one absolute value:

Isolate the absolute value on one side of the equation.

Is the number on the other side of the equation negative?
If you answered yes, then the equation has no solution. If you answered
no, then go on to step 3.

Write two equations without absolute values. The first equation
will set the quantity inside the bars equal to the number on the other
side of the equal sign; the second equation will set the quantity inside
the bars equal to the opposite of the number on the other side.

Solve the two equations.

Follow these steps to solve an absolute value equality
which contains two absolute values (one on each side of the equation):

Write two equations without absolute values. The first
equation will set the quantity inside the bars on the left side equal to
the quantity inside the bars on the right side. The second equation
will set the quantity inside the bars on the left side equal to the opposite
of the quantity inside the bars on the right side.

Solve the two equations.

Let's look at some examples.

Example 1: Solve |2x - 1| + 3 = 6

Step 1:Isolate
the absolute value

|2x - 1| + 3 = 6

|2x - 1| = 3

Step 2:Is
the number on the other side of the equation negative?

No, itís a positive number, 3, so continue on to
step 3

Step 3:Write
two equations without absolute value bars

2x - 1 = 3

2x - 1 = -3

Step 4:Solve
both equations

2x - 1 = 3

2x = 4

x = 2

2x - 1 = -3

2x = -2

x = -1

Example 2: Solve |3x - 6| - 9 = -3

Step 1:Isolate
the absolute value

|3x - 6| - 9 = -3

|3x - 6| = 6

Step 2:Is
the number on the other side of the equation negative?

No, itís a positive number, 6, so continue on to
step 3

Step 3:Write
two equations without absolute value bars

3x - 6 = 6

3x - 6 = -6

Step 4:Solve
both equations

3x - 6 = 6

3x = 12

x = 4

3x - 6 = -6

3x = 0

x = 0

Example 3: Solve |5x + 4| + 10 = 2

Step 1:Isolate
the absolute value

|5x + 4| + 10 = 2

|5x + 4| = -8

Step 2:Is
the number on the other side of the equation negative?

Yes, itís a negative number, -8. There is no solution
to this problem.

Example 4: Solve |x - 7| = |2x - 2|

Step 1:Write
two equations without absolute value bars

x - 7 = 2x - 2

x - 7 = -(2x - 2)

Step 4:Solve
both equations

x - 7 = 2x - 2

-x - 7 = -2

-x = 5

x = -5

x - 7 = -2x + 2

3x - 7= 2

3x = 9

x = 3

Example 5: Solve |x - 3| = |x + 2|

Step 1:Write
two equations without absolute value bars

x - 3 = x + 2

x - 3 = -(x + 2)

Step 4:Solve
both equations

x - 3 = x + 2

- 3 = -2

false statement

No solution from this equation

x - 3 = -x - 2

2x - 3= -2

2x = 1

x = 1/2

So the only solution to this problem is x = 1/2

Example 6: Solve |x - 3| = |3 - x|

Step 1:Write
two equations without absolute value bars

x - 3 = 3 - x

x - 3 = -(3 - x)

Step 4:Solve
both equations

x - 3 = 3 - x

2x - 3 = 3

2x = 6

x = 3

x - 3 = -(3 - x)

x - 3= -3 + x

-3 = -3

All real numbers are solutions to this equation

Since 3 is included in the set of real numbers,
we will just say that the solution to this equation is All Real Numbers